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Question

If f(x)=(logcotxtan x)(logtanxcot x)1+tan1(x(4x2)) then f'(0) is equal to

A
2
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B
2
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C
12
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D
0
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Solution

The correct option is C 12
f(x)=(logcotxtan x)(logtan xcot x)+tan1(x(4x2))
f(x)=1+tan1(x(4x2))
Now Put x=2sinθ, we get
f(x)=1+tan1(2sinθ44sin2θ)f(x)=1+tan1(2sinθ2cosθ)f(x)=1+tan1(tanθ)f(x)=1+θf(x)=1+sin1(x2)f(x)=0+11(x2)212f(0)=12

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